# How do you solve 10^(4x -1) = 1000 and find any extraneous solutions?

May 5, 2018

The only solution is $x = 1$.

#### Explanation:

Rewrite $1000$ as ${10}^{3}$:

${10}^{4 x - 1} = 1000$

${10}^{4 x - 1} = {10}^{3}$

Now, since the bases are the same, the exponents must be equal to each other:

${10}^{\textcolor{red}{4 x - 1}} = {10}^{\textcolor{red}{3}}$

$4 x - 1 = 3$

$4 x = 4$

$x = 1$

This is the only solution. We can verify it by plugging it back into the original equation:

${10}^{4 \left(1\right) - 1} = 1000$

${10}^{4 - 1} = 1000$

${10}^{3} = 1000$

$1000 = 1000$

Hope this helped!